Numerical differentiation and parameter estimation in higher-order linear stochastic systems

For a linear time-invariant system of order d greater than or equal to 2 with a white noise disturbance, the input and the output are assumed to be sampled at regular time intervals. Using only these observations, some approximate values of the first d - 1 derivatives are obtained by a numerical differentiation scheme, and the unknown system parameters are estimated by a discretization of the continuous-time least-squares formulas, These parameter estimates have an error which does not approach zero as the sampling interval approaches zero, This asymptotic error is shown to be associated with the inconsistency of the quadratic variation estimate of the white noise local variance based on the sampled observations. The use of an explicit correction term in the least-squares estimates or the use of some special numerical differentiation formulas eliminates the error in the estimates.